theorem
  x in H * A iff ex g1,g2 st x = g1 * g2 & g1 in H & g2 in A
proof
  thus x in H * A implies ex g1,g2 st x = g1 * g2 & g1 in H & g2 in A
  proof
    assume x in H * A;
    then consider g1,g2 such that
A1: x = g1 * g2 and
A2: g1 in carr H and
A3: g2 in A;
    g1 in H by A2;
    hence thesis by A1,A3;
  end;
  given g1,g2 such that
A4: x = g1 * g2 and
A5: g1 in H and
A6: g2 in A;
  g1 in carr H by A5;
  hence thesis by A4,A6;
end;
